Numerical simulation of 3D Darcy–Forchheimer fluid flow with the energy and mass transfer over an irregular permeable surface

The Jeffrey fluid model is capable of accurately characterizing the stress relaxation behavior of non-Newtonian fluids, which a normal viscous fluid model is unable to perform. The primary objective of this paper is to provide a comprehensive investigation into the effects of MHD and thermal radiation on the 3D Jeffery fluid flow over a permeable irregular stretching surface. The consequences of the Darcy effect, variable thickness and chemical reaction are also considered. The phenomena have been modeled as a nonlinear system of PDEs. Using similarity substitution, the modeled equations are reduced to a dimensionless system of ODEs. The parametric continuation method (PCM) is used to determine the numerical solution to the obtained sets of nonlinear differential equations. The impact of physical parameters on temperature, velocity and mass profiles are presented through Figures and Tables. It has been noticed that the energy profile magnifies with the increment of porosity term, thermal radiation and heat source term, while diminishing with the flourishing upshot of power index and Deborah number. Furthermore, the porosity term and wall thickness parameter enhance the skin friction.

www.nature.com/scientificreports/ of the magnetic field, Marangoni factor, and unsteadiness component decreases the fluid velocity. Some advanced research have been reported by [11][12][13][14] . Non-Newtonian fluids come in a wide variety of types, each with its own set of characteristics. Non-Newtonian fluids are being examined by scientists and researchers due to a broad range of implementations, like drug companies, fiber new tech, cables sealant, food items, crystal growth, psychology and many more. Jeffrey fluid is the most well-known and easiest-to-understand. The Jeffrey fluid parameter and the time retardation parameter elevates this fluid to the top of the non-Newtonian fluids list 15 . Ullah et al. 16 evaluated the Jeffrey fluid flow across porous horizontal sheet. The velocity of an unsteady Jeffrey fluid flow over an inestimable plane permeable plate is inspected by Algehyne et al. 17 . The results reveal that as the magnetic parameter, the ratio of retardation and relaxation times and Jeffrey fluid parameter increase, the fluid velocity decreases. In the response to an ambient magnetic field, Ali et al. 18 calculated the upshot of energy conduction on the flow of a Jeffery fluid with immersed NPs through a dynamic flexible substrate. Alrabaiah et al. 19 inspected the peristaltic transmission of MHD Jeffery fluid flow through channel. Saleem et al. evaluated MHD Jeffrey fluid flows with mass and energy transport on an indefinitely circulating inverted cone. 20 . Azlina et al. 21 proposed a numerical calculation of the MHD Jeffrey fluid flow through plates in a translucent sheet. Bilal et al. 22 investigate the 2D Jeffrey fluid flow across a continuously extending disc. Kumar et al. 23 discussed the influence of an applied magnetics flux on an irregular 2D Jeffrey fluid flow. A theoretical investigation is carried out by Yadav et al. 24 to determine the upshot of a magnetic flux and mixed convection on the Jeffrey fluid flow. The results show that increasing the Jeffrey fluid parameter reduces system stability while increasing magnetic field parameters has the reverse effect. Recently many researchers have worked on this topic [25][26][27][28] .
The MHD flow plays a vital role in manufacturing heavy machinery, astrophysics, electrical power generation solar power equipment, space vehicle and many other fields. Kumar et al. 29 explore the thermal energy transference in a HNF flow through an extending cylinder while considering magnetic dipoles. Nanoliquid flow across curved stretched sheets is studied numerically by Dhananjaya et al. 30 to determine the effect of magnetic fields on Casson nanoliquid flow. The findings indicated that enhancing the curvature parameter positively affects the velocity profile, but that it has the opposite impact on the thermal gradient. Chu et al. 31 scrutinise Maxwell nano liquid's radiative flow along with a cylinder by taking into consideration the magnetic effect. The fluid flow and temperature fluctuations of nanofluid flow with the Hall upshot are discussed by Acharya et al. 32 . A moving plate with Joule heating is used to demonstrate Magnetohydrodynamic hybrid nanofluid flow with temperature distribution is solved numerically by Lv et al. 33 . Kodi et al. 34 presented an analytical assessment of Casson fluid flows with heat and mass transmit. This analysis revealed that intensifying the Newtonian heating effect shrinks heat transport at the plate surface. The influence of a porous surface and magnetic flux on the Jeffery fluid flow has been reported by Abdelhameed 35 . Ellahi et al. 36 investigated the upshots of MHD and velocity slip on sliding flat plate. The obtained outcomes exposed that the velocity contour improves for different values of the slip variable. Recently, a large number of studies have been reported by the implying magnetic effect on the fluid flow [37][38][39][40] .
The Jeffrey fluid model effectively describes the stress relaxation behavior of non-Newtonian fluids, which is something that the standard viscous fluid model can't. The Jeffrey fluid model may accurately explain a class of non-Newtonian fluids. The main purpose of this research is to look into the impact of MHD and thermal radiation on the 3D Jeffery fluid flow over an irregular stretching surface. The Darcy effect, varying thickness, and chemical reaction are all taken into account. The results are obtained through computational strategy PCM.

Mathematical formulation
The influence of a tridimensional steady MHD Jeffery fluid flow on an irregular surface immersed in an absorbent medium is considered. Figure 1 described a schematic description of the model. The magnetic effect B is imposed in the z-direction. When the fluid is stationary at t = 0, the sheet is impulsively stretched in the x and y directions with velocities u w and v w . The effects of solar radiation on the sheet's surface as well as chemical reaction are considered. Under the above description, the principal equations are expressed as 14 : Here (u, v, w) determine the velocity factors in x, y and z direction. k p is the permeability of the porous medium, k f is the thermal conductivity, T is the temperature of the fluid, ν is the kinematic viscosity, Q the heat absorption/ (1) www.nature.com/scientificreports/ generation term, F is the non-uniform inertia factor, where, C b is the drag coefficient. D is the molecular diffusivity and 1 , 2 is the period of relaxation and time retardation respectively. The boundary conditions are 14,41 : where In the above equation, we supposed as n = 1 (i.e., n = 1 denotes the surface shape to flat sheet). Where, n > 1 and n < 1 are yields to surface curviness, inner convex and outer convex due to reduction and increment of wall thicknesses respectively. T 0 , T ∞ reference atmospheric liquid temperature f 1 specify the Maxwell coefficient, b shows the thermal adaptation coefficient, ζ 1 , ζ 2 are the constant number, specific heat ratio.

Similarity transformation
The similarity variables are: By applying the above similarity transformation, Eq. (1) is identically satisfied while Eq. (2-5) take the form as: Here, D and M F is the Deborah number and magnetic field, Hs is the absorption & generation term, P 0 is the porosity factor, R is the thermal radiation, Pr and Sc is the Prandtl and Schmidt numbers, Fr is the Darcy Forchhemier term, C r is the chemical reaction and wall thickness factor. Mathematically we have The friction factor towards x and y direction are: where Here the physical quantities are: The skin friction, heat and mass allocation expression are as follows: Here Re = �U w ν f is the Reynold's number. (8) Re 0.5 y Cf y =

Numerical solution
The basic methodology steps of PCM approach are as follow 17,42-46 : Step 1: simplification to 1st order ODE. with the corresponding boundary conditions.
Step 3: solving the Cauchy principal. Numerical implicit scheme is employed for the above modeled equations, which is defined as below: Finally, we get:

Results and discussion
The section revealed the physics behind each figure and table. The subsequent trends have been observed:    www.nature.com/scientificreports/ with the porosity parameter effect, which resists the fluid flow, so causes the reduction in the velocity outline. Figures 4 and 5 reported that the rising frequency of both constraints magnetic field and Darcy Forchhemier effect deduce the velocity distribution. Because the opposing force, which is created due to magnetic effect, resist the flow field, as a result such trend observed. Figures 6 and 7 described that the impact of local Deborah number   www.nature.com/scientificreports/ and wall thickness parameter augmented the velocity field. Figure 8 displays that the upshot of thermal relaxation term, reduce the energy profile.
Energy profile. Figures 9, 10, 11, 12, 13, 14 explained the appearance of energy contour θ (η) versus the variation of porosity term P 0 , power-law index n, thermal radiation R, heat source term Hs, Deborah number        Error analysis. In Fig. 18, we performed the error analysis, to ensure that our results are accurate up to the lowest residual error scale. Until evaluating and providing physical forecasts, we analyze an error to determine the accuracy of the proposed method. Tables 1 and 2 illustrated the statistical outcomes for skin friction, Nusselt and Sherwood number versus several physical constraints respectively. Table 3 highlighted the comparative assessment of the present results versus the existing works. The results of Table 3 verify the accuracy of the current analysis.  www.nature.com/scientificreports/

Conclusion
We have numerically analyzed the energy conveyance through Jeffery fluid flow over an irregular extensible sheet with a porous medium. The consequences of the Darcy effect, variable thickness and chemical reaction are also considered. The phenomena have been modeled as a system of PDEs. Using similarity substitution, the modeled equations are reduced to a dimensionless system of ODEs. The computational technique is used to determine the numerical solution to the obtained sets of nonlinear differential equations. The key conclusions are: • The velocity profiles f ′ (η), g ′ (η) both decline with the increment of porosity term, magnetic field, Darcy Forchhemier and thermal relaxation factor while augmented with the flourishing upshot of power index and local Deborah number. • The energy profile θ(η) magnifies with the increment of porosity term, thermal radiation and heat source term, while diminishing with the flourishing upshot of power index and Deborah number. • The mass transfer profiles reduce with the rising upshot of C R , Deborah number and Schmidt number.  Table 3. Relative evaluation of current results with the available literature for −f ′′ (0). n Reddy et al. 41 Khader et al. 47 Khan et al. 48